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Cluster points of a sequence $a_n$

0

$begingroup$

Suppose that there is a sequence $a_n$ with the below subsequnces:

a) $a_n = fracn+2+1n^2 + n + 1$ when $n$ is odd.

b) $a_n = fracsinnn$ when $n$ is mulptiple of $4$

c) $a_n = frac1n - 1$ when $n$ is even but not multiple of $4$

Which are the cluster points?
The solution is $A=-1,0,1$ but I don't understand why.

My solution:

a) $a_n = fracn+2+1n^2+n+1 = frac1 + frac2n + frac1nn+1+frac1n rightarrow 0$

b) $ frac-1n leq fracsinnn leq frac1n ... rightarrow 0$

c) $ a_n = frac1n - 1 rightarrow -1$

So, $A = -1,0 $

How did they find the cluster point $1$ ?

real-analysis

share|cite|improve this question

edited Mar 23 at 16:30

Dimitris Dimitriadis

asked Mar 23 at 16:18

Dimitris DimitriadisDimitris Dimitriadis

568

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  The subsequence c is $a_n = 1/n - n$ or $a_n = 1/n - 1$ like in your solution?
  $endgroup$
  – The Student
  Mar 23 at 16:29
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I edited it. Thank you
  $endgroup$
  – Dimitris Dimitriadis
  Mar 23 at 16:30
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  The subsequence c converges to -1. So, why -1 isn't a cluster point ?
  $endgroup$
  – Dimitris Dimitriadis
  Mar 23 at 16:41
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Sorry, is 1 instead of -1
  $endgroup$
  – The Student
  Mar 23 at 16:42
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Ohh ok !! I have the same opinion about this. It is an exercise of a book that gives this solution. I mean $A = -1,0,1 $
  $endgroup$
  – Dimitris Dimitriadis
  Mar 23 at 16:43
  
  

  
  
  
  

 | 
show 2 more comments

0

$begingroup$

Suppose that there is a sequence $a_n$ with the below subsequnces:

a) $a_n = fracn+2+1n^2 + n + 1$ when $n$ is odd.

b) $a_n = fracsinnn$ when $n$ is mulptiple of $4$

c) $a_n = frac1n - 1$ when $n$ is even but not multiple of $4$

Which are the cluster points?
The solution is $A=-1,0,1$ but I don't understand why.

My solution:

a) $a_n = fracn+2+1n^2+n+1 = frac1 + frac2n + frac1nn+1+frac1n rightarrow 0$

b) $ frac-1n leq fracsinnn leq frac1n ... rightarrow 0$

c) $ a_n = frac1n - 1 rightarrow -1$

So, $A = -1,0 $

How did they find the cluster point $1$ ?

real-analysis

share|cite|improve this question

edited Mar 23 at 16:30

Dimitris Dimitriadis

asked Mar 23 at 16:18

Dimitris DimitriadisDimitris Dimitriadis

568

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  The subsequence c is $a_n = 1/n - n$ or $a_n = 1/n - 1$ like in your solution?
  $endgroup$
  – The Student
  Mar 23 at 16:29
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I edited it. Thank you
  $endgroup$
  – Dimitris Dimitriadis
  Mar 23 at 16:30
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  The subsequence c converges to -1. So, why -1 isn't a cluster point ?
  $endgroup$
  – Dimitris Dimitriadis
  Mar 23 at 16:41
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Sorry, is 1 instead of -1
  $endgroup$
  – The Student
  Mar 23 at 16:42
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Ohh ok !! I have the same opinion about this. It is an exercise of a book that gives this solution. I mean $A = -1,0,1 $
  $endgroup$
  – Dimitris Dimitriadis
  Mar 23 at 16:43
  
  

  
  
  
  

 | 
show 2 more comments

$begingroup$

Suppose that there is a sequence $a_n$ with the below subsequnces:

a) $a_n = fracn+2+1n^2 + n + 1$ when $n$ is odd.

b) $a_n = fracsinnn$ when $n$ is mulptiple of $4$

c) $a_n = frac1n - 1$ when $n$ is even but not multiple of $4$

Which are the cluster points?
The solution is $A=-1,0,1$ but I don't understand why.

My solution:

a) $a_n = fracn+2+1n^2+n+1 = frac1 + frac2n + frac1nn+1+frac1n rightarrow 0$

b) $ frac-1n leq fracsinnn leq frac1n ... rightarrow 0$

c) $ a_n = frac1n - 1 rightarrow -1$

So, $A = -1,0 $

How did they find the cluster point $1$ ?

real-analysis

share|cite|improve this question

edited Mar 23 at 16:30

Dimitris Dimitriadis

asked Mar 23 at 16:18

Dimitris DimitriadisDimitris Dimitriadis

568

$endgroup$

share|cite|improve this question

edited Mar 23 at 16:30

Dimitris Dimitriadis

asked Mar 23 at 16:18

Dimitris DimitriadisDimitris Dimitriadis

568

share|cite|improve this question

edited Mar 23 at 16:30

Dimitris Dimitriadis

asked Mar 23 at 16:18

Dimitris DimitriadisDimitris Dimitriadis

568

asked Mar 23 at 16:18

Dimitris DimitriadisDimitris Dimitriadis

568

asked Mar 23 at 16:18

Dimitris DimitriadisDimitris Dimitriadis

568